When you ask a mountaineer why he is doing what he is doing a possible answer is: “Because it’s there“. Are mathematicians any different? Our motivation for doing mathematics also stems from the fact that there are still significant mathematical mountains. For example, the holy grail of mathematics, the Riemann Hypothesis is considered hard, interesting and a solution is worth a million dollar. Most likely new methods and ideas have to be developed and they will most certainly contain surprises for the experts. Several fields in mathematics would take a gigantic leap forward.
However, the way we do mathematics or the way we think about mathematics will not be changed. Whatever the solution will be, we do not expect it to be as staggering as quantum theory was for physics or as breathtaking as evolution theory was for biology. This is certainly good news if one thinks of mathematics as a dogmatic science. On the other hand, there might be interesting insights if we only just bend our preconceptions on the axiomatic approach or towards circularity a little. If successful, we might change our understanding of what mathematics actually is and how it should be done.
Are there widely accepted scientific problems for which we so far have no mathematical description or solution? Problems that are so hard, that maybe they will change the way we do mathematics? Problems that give us the impression that we only know what mathematics was in the past and that we shall never know what mathematics will look like in the future?
Without claim of completeness:
In biology, evolution captures the alteration of populations consisting of reproducing individuals and their environment over time by properties of the individuals. Well known concepts such as natural selection and “survival of the fittest” are justified by a plethora of observations and therefore the theory is a cornerstone of modern science. 150 years of research and there still is no comprehensive mathematical treatment of evolution. Why is this so hard? I have no answer to this, although two observations come to mind. 20th century mathematics is usually presented as a typed theory and thus promotes populations over individuals. From that viewpoint, evolution seems to consist of special cases only rather than to contain some abstract, mathematical content. Additionally, there are obvious circularities in the considered notions population/environment and survival/fitness. The biological content seems to be rather in these circles than in some mathematical bottom-up framework of axioms and theorems.
In economics, markets are places where agents satisfy their demand for goods by trading them. For more than 100 years we have markets as objects under scientific consideration. So far, we do not have a time evolution for prices and demands maybe even in terms of actions taken by agents and preferably extending existing results in microeconomics. What is the difficulty? Maybe the following observation sheds some light on the situation. It seems that correctly predicted prices lead to arbitrages which then change the behavior/demand of the agents. That circle seems to encode the basic essentials of economics. Theories with assumptions on either side, the prices or the demand, seem to assume away the problem and are then only loosely connected to observations. This reminds on games with incomplete information. Making assumptions on opponents strategies greatly simplifies the search for ones own optimal strategy. However, justifying these assumptions is the hard part with not much mathematics involved so far.
In physics, cosmology studies the dynamics and the evolution of the universe as a whole. By this definition an observer is part of the universe and thus ‘laws’ governing this universe put restrictions on what ‘observation’ actually is. On the other hand, the observer develops models describing the universe and thus himself as part of this universe. This circular observer/universe dependence is related to anthropic principles. The problem now consists in developing a correct argument to refute models essentially just by applying the observational fact of the existence of the observer. This is hard since the ‘logic’ an observer can use to reason about the universe is also part of the universe and might change if the universe is changed. As long as we do not know what ‘observation’ means it is not enough to exclude possible models just by the non-existence of a certain type of observer (e.g. carbon based life form), since there could be other observers experiencing their observations as e.g. we do ours and thus different models could still be ‘isomorphic as observed universes’ (whatever that means).
IV. Foundation of Physics
In physics, the outcome of conducted experiments and of developed theories is compared according to established rules. If the comparison meets certain standards the theory is called physical and is said to describe reality. The outcome of an experiment is usually described in terms of fundamental units of measurement. Different social communities use different units. A widely accepted standard is SI. Within SI the unit of ‘mass’ is kilogram, which is defined by reference to a certain pile of platinum-iridium alloy stored at a certain place in France. This procedure puts severe limits to the accuracy of measurements (as of 2010 this is one part in 10^8). The ultimate goal therefore is to liberate the SI system’s dependency on this definition by developing a practical realization of the kilogram that can be reproduced in different laboratories by following a written specification. The units of measure in such a practical realization would have their magnitudes precisely defined and expressed in terms of fundamental physical constants. Why has nobody done that so far? The problem seems to be to describe the foundations of physics in physical terms. That circularity seems to be hard to overcome. Moreover, the above argumentation is not only valid for ‘mass’, but for any other physical quantity and one would end up with a huge network of interdependent ‘written specifications’.
We know that there is life, that there is intelligence and that there is conciousness. Mathematics has little more to say about these. What is the problem? We might not have enough data, different intelligent life forms, to extract an abstract theory. However, we also might face a semantical problem. In 20th century mathematics we know of two ways to introduce notions. First, by description, like set or element. These notions are given by their relation to other described notions in a huge semantical circle. No further justification for them is given. Second, by definition, like function or number. Defined notions come together with a substitution process and can, at least in principle, be eliminated from the theory. The notions in the title so far withstood any attempts to being defined. It seems that these notions ’emerge’ by some sort of ‘limit’ procedure from other notions. Of course, this is only speculation and anytime soon a reasonable definition of ‘life’, ‘intelligence’ and ‘conciousness’ might be found. If not, however, we might take that as an indication that mathematics still has potential rather than limits.